I will be very grateful if someone could provide me the solution of the following question:
(a) Find the solution of the equation
. . . . .\(\displaystyle \left(1\, +\, {x_1}^2\right)u_{x1}\, +\, u_{x2}\, =\, u^2\)
satisfying the condition
. . . . .\(\displaystyle u\left(0,\, x_2\right)\, =\, g\left(x_2\right)\)
where \(\displaystyle g\, :\, \mathbb{R}\, \rightarrow\, \mathbb{R}\) is a smooth positive function.
(b) Find the largest domain on which the solution is defined in the case that \(\displaystyle g\, \equiv\, 1.\)
(a) Find the solution of the equation
. . . . .\(\displaystyle \left(1\, +\, {x_1}^2\right)u_{x1}\, +\, u_{x2}\, =\, u^2\)
satisfying the condition
. . . . .\(\displaystyle u\left(0,\, x_2\right)\, =\, g\left(x_2\right)\)
where \(\displaystyle g\, :\, \mathbb{R}\, \rightarrow\, \mathbb{R}\) is a smooth positive function.
(b) Find the largest domain on which the solution is defined in the case that \(\displaystyle g\, \equiv\, 1.\)
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